Simplify the following expression and state the condition under which the simplification is valid: $a = \dfrac{q^2 - 5q - 24}{q^2 - 7q - 8}$
Explanation: First factor the expressions in the numerator and denominator. $ \dfrac{q^2 - 5q - 24}{q^2 - 7q - 8} = \dfrac{(q + 3)(q - 8)}{(q + 1)(q - 8)} $ Notice that the term $(q - 8)$ appears in both the numerator and denominator. Dividing both the numerator and denominator by $(q - 8)$ gives: $a = \dfrac{q + 3}{q + 1}$ Since we divided by $(q - 8)$, $q \neq 8$. $a = \dfrac{q + 3}{q + 1}; \space q \neq 8$